Modeling the (proto)neutron star crust : toward a controlled estimation of uncertainties

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Thomas Carreau

To cite this version:

Pour obtenir le diplôme de doctorat

Spécialité PHYSIQUE

Préparée au sein de l'Université de Caen Normandie

Μοdeling the (prοtο)neutrοn star crust: tοward a cοntrοlled

estimatiοn οf uncertainties

Présentée et soutenue par

Thomas CARREAU

Thèse soutenue publiquement le 25/09/2020 devant le jury composé de

Mme MICAELA OERTEL Directeur de recherche au CNRS, LUTH,_{CNRS/Observatoire de Paris} Rapporteur du jury Mme CONSTANCA PROVIDENCIA Professeur, Université de Coïmbra Rapporteur du jury M. NICOLAS CHAMEL Maître de recherches, Fonds de la Recherche_{Scientifique} Membre du jury Mme ANTHEA FANTINA Chargé de recherche au CNRS, 14 GANIL de_CAEN Membre du jury M. JEROME MARGUERON Directeur de recherche au CNRS, Institut de_{Physique des 2 Infinis} Président du jury Mme FRANCESCA GULMINELLI Professeur des universités, Université Caen_Normandie Directeur de thèse

Je souhaite tout d’abord remercier le directeur du LPC Caen, Gilles Ban, de m’avoir accueilli au sein du laboratoire.

I would like to thank the jury members Nicolas Chamel, Anthea F. Fantina, Jérôme Margueron, Micaela Oertel, and Constança Providência who carefully read this manuscript. I have appreciated the referees Micaela Oertel and Constança Providên-cia having shown interest in my work during the defense. I thank them and Nicolas Chamel for their useful comments and corrections.

Je tiens à exprimer tout mon estime à Francesca par qui j’ai eu la chance d’être encadré pendant trois ans et demi, en tenant compte du stage de deuxième année de master. J’ai énormément appris de nos nombreux échanges. Francesca a toujours pris le temps de répondre à mes questions et m’a accordé une grande liberté concernant l’encadrement de stages, l’enseignement, la participation aux conférences ou encore la médiation scientifique. Pour toutes ces raisons, je la remercie sincèrement.

Je remercie vivement Anthea pour les discussions intéressantes que nous avons pu avoir et d’avoir été ma guide à Bruxelles lors de nos visites rendues à Nicolas. Je remercie par ailleurs Nicolas pour son accueil chaleureux à l’ULB et ses conseils. J’ai grandement apprécié travailler avec Anthea et Nicolas.

Je remercie également Jérôme avec qui j’ai eu la chance de collaborer. J’ai apprécié nos échanges lors de nos rencontres en conférence ou à l’IP2I.

Je souhaite remercier l’ensemble des membres du LPC de contribuer à l’ambiance conviviale qui reigne au sein du laboratoire. Y Préparer ma thèse pendant ces trois ans fut un grand plaisir.

Un grand merci aux thésards (et ex-thésards !) ainsi qu’aux postdocs du labora-toire pour tous les très bons moments passés ensemble au labo et lors des soirées. Merci à Daniel d’avoir fait le déplacement depuis Cherbourg pour venir assister à ma soute-nance. Joël, j’espère que je ne te manquerai pas trop lors des trajets quotidiens∗_{. Bon}

courage à notre rugbyman Edgar pour la dernière ligne droite ! Je souhaite également bon courage à Cyril et Joël ainsi qu’à la jeune génération Alexandre, Chloé, Savitri et William pour la suite jusqu’à la soutenance. Yoann, j’ai adoré être ton collègue de bureau†_{. Merci d’avoir contrôlé la caméra le jour de la soutenance.}

∗_{J’espère que tu sauras faire la différence entre un panneau stop et un feu rouge à l’avenir !} †_{à mi-temps !}

Valérian, on a fait toutes nos années d’études ensemble et on va maintenant être séparés par plus que quelques mètres pendant les années à venir. Nos discussions et après-midi musique vont me manquer‡_{. Bonne chance à Marie et toi en Allemagne,}

on passera vous voir !

Merci à mon ami d’enfance Alexis qui m’a suivi tout le long de ce parcours (depuis la maternelle !).

Enfin, je remercie naturellement mon petit frère Alexis et mes parents de m’avoir toujours soutenu dans mes choix. Pour la même raison, je tiens également à remercier Catherine.

Ces derniers mots sont destinés à Manon qui m’accompagne depuis le début. Merci pour ton soutien et tes encouragements sans failles, particulièrement durant ces derniers mois où nous étions confinés tous les deux et que je rédigeais. Tu es merveilleuse et je suis heureux de partager cette réussite avec toi.

List of Figures ix

List of Tables xi

General introduction 1

1 Structure and equation of state of cold nonaccreting neutron stars 9

1.1 Ground state of the outer crust . . . 11

1.1.1 Wigner-Seitz cell energetics . . . 11

1.1.1.1 Nuclear mass tables . . . 12

1.1.1.2 Relativistic electron gas . . . 12

1.1.2 The BPS model . . . 13

1.1.3 Equilibrium composition and equation of state . . . 15

1.2 Ground state of the inner crust . . . 19

1.2.1 Modeling the nuclear energy . . . 20

1.2.1.1 Metamodeling of homogeneous nuclear matter . . . 22

1.2.1.2 From homogeneous nuclear matter to finite nuclei in the CLD approximation . . . 27 1.2.2 Variational formalism . . . 31 1.2.3 Results . . . 35 1.2.3.1 Numerical code . . . 35 1.2.3.2 Equilibrium composition . . . 36 1.2.3.3 Equation of state . . . 38

1.2.4 Strutinsky shell corrections to the CLD energy . . . 39

1.2.5 Nonspherical pasta phases . . . 42

1.2.6 Crust-core transition from the crust side . . . 45

1.3 Matter of the core . . . 47

1.3.1 Outer core: homogeneous npeµ matter . . . 48

1.3.1.1 Variational equations . . . 49

1.3.1.2 Equilibrium composition and equation of state . . . 50

1.3.2 Inner core . . . 51

1.4 Unified metamodeling of the equation of state . . . 52

2 Bayesian inference of neutron star observables 57

2.1 From the equation of state to neutron star observables . . . 59

2.1.1 Masses and radii . . . 59

2.1.1.1 Mass-radius relation . . . 59

2.1.1.2 Crust thickness and mass . . . 62

2.1.2 Moment of inertia within the slow rotation approximation . . . 63

2.1.2.1 Total moment of inertia and fraction contained in the crust . . . 64

2.1.2.2 Connection to pulsar glitches . . . 66

2.1.3 Tidal deformability . . . 68

2.2 Bayesian determination of the equation of state parameters . . . 70

2.2.1 Principle of Bayesian inference . . . 70

2.2.2 Prior distribution of equation of state parameters . . . 72

2.2.2.1 Flat prior compatible with empirical constraints . . . . 72

2.2.2.2 Sensitivity analysis of the crust-core transition point . 73 2.2.3 Determination of the likelihood function . . . 76

2.2.3.1 Constraints on nuclear physics observables . . . 76

2.2.3.2 Physical requirements and constraints on neutron star observables . . . 78

2.2.4 Posterior distribution of equation of state parameters . . . 78

2.2.4.1 Marginalized one-parameter posterior distributions of empirical parameters . . . 79

2.2.4.2 Correlations among empirical parameters . . . 82

2.3 General predictions for neutron star observables . . . 84

2.3.1 Global properties: confrontation with popular models and the GW170817 event . . . 84

2.3.1.1 Equation of state . . . 85

2.3.1.2 Masses and radii . . . 86

2.3.1.3 Tidal deformability . . . 88

2.3.2 Bayesian analysis of the crust-core transition . . . 91

2.3.3 Pulsar glitches: answering the question “Is the crust enough?” . 96 2.4 Conclusions . . . 102

3 Crystallization of the crust of protoneutron stars 105 3.1 Model of the crust at finite temperature . . . 107

3.1.1 One-component Coulomb plasma approximation . . . 107

3.1.1.1 OCP in a liquid phase . . . 108

3.1.1.2 OCP in a solid phase . . . 109

3.1.1.3 Crystallization of a OCP . . . 110

3.1.2 Multicomponent Coulomb plasma in a liquid phase . . . 112

3.1.2.1 Nuclear statistical equilibrium . . . 112

3.1.2.2 Evaluation of the chemical potentials . . . 114

3.1.2.3 Evaluation of the rearrangement term . . . 115

3.1.3 Nuclear free energy functional in the free neutron regime . . . . 116

3.1.3.2 Energetics of clusters at finite temperature in the CLD

approximation . . . 120

3.2 Study of the outer crust at crystallization . . . 120

3.2.1 Crystallization temperature . . . 122

3.2.2 Equilibrium composition . . . 122

3.2.3 Impurity parameter . . . 126

3.2.4 Abundancies of odd nuclei . . . 126

3.3 Study of the inner crust at crystallization . . . 129

3.3.1 Influence of shell effects in the OCP approximation . . . 130

3.3.1.1 Temperature dependence of shell corrections . . . 130

3.3.1.2 Equilibrium composition of the OCP at crystallization for modern BSk functionals . . . 133

3.3.2 MCP results . . . 136

3.3.2.1 Equilibrium composition of the MCP . . . 137

3.3.2.2 Impurity parameter . . . 139

3.4 Conclusions . . . 140

General conclusions and outlooks 143 A Energy density of a relativistic electron gas 147 B Neutron and proton chemical potentials in the metamodel 149 C Partie en français 151 C.1 Introduction générale . . . 151

C.2 Structure et équation d’état des étoiles à neutrons froides non accrétantes157 C.3 Inférence bayésienne des observables des étoiles à neutrons . . . 158

C.4 Cristallisation de la croûte des protoétoiles à neutrons . . . 160

C.5 Conclusions générales et perspectives . . . 162

1 Schematic representation of the inside of a neutron star . . . 2

2 Masses measured from pulsar timing . . . 4

1.1 Ground-state composition versus baryon density in the outer crust . . . 18

1.2 Pressure versus baryon density in the outer crust . . . 19

1.3 r-cluster and e-cluster representations . . . 20

1.4 Accuracy of the metamodeling technique for SLy4 . . . 26

1.5 Fit of surface and curvature parameters . . . 29

1.6 Surface plus curvature energy per surface nucleon versus asymmetry . . 30

1.7 Ground-state composition versus baryon density in the inner crust . . . 36

1.8 Energy per nucleon and pressure versus baryon density in the inner crust, and symmetry energy at subsaturation densities . . . 38

1.9 Perturbative implementation of proton shell corrections for BSk24 . . . 40

1.10 Equilibrium value of Z versus baryon density in the inner crust with Strutinsky shell corrections . . . 41

1.11 Pasta phases in the inner crust . . . 44

1.12 Crust-core transition density versus surface parameter p for SLy4 . . . 46

1.13 Crust-core transition density and pressure for several interactions . . . 48

1.14 Equilibrium composition, pressure, and symmetry energy versus baryon density in the outer core . . . 50

1.15 Unified metamodeling of the equation of state . . . 53

2.1 Mass-radius relation for several popular equations of state . . . 61

2.2 Crust thickness and mass versus neutron star mass for several popular equations of state . . . 63

2.3 Moment of inertia and fraction of crust moment of inertia versus neutron star mass for several popular equations of state . . . 65

2.4 Tidal Love number and dimensionless tidal deformability versus neutron star mass for several popular equations of state . . . 69

2.5 Sensitivity analysis of the crust-core transition point for BSk24 and BSk22 75 2.6 Energy per nucleon and pressure of nuclear matter versus density from chiral effective field theory caclulations . . . 77

2.7 Marginalized posteriors for isoscalar empirical parameters assuming dif-ferent filters . . . 80

2.8 Marginalized posteriors for isovector empirical parameters assuming dif-ferent filters . . . 81

2.9 Correlations among empirical parameters for the sets passing through the low-density filter . . . 82 2.10 Correlations among empirical parameters for the sets passing through

both low-density and high-density filters . . . 83 2.11 Posterior equation of state confronted with popular models and the

GW170817 event . . . 85 2.12 Posterior mass-radius relation confronted with popular models and the

GW170817 event . . . 87 2.13 Posterior Λ1 − Λ2 relation confronted with popular models and the

GW170817 event . . . 89 2.14 Marginalized posterior probability distribution for the dimensionless

tidal deformability of a 1.4M⊙ neutron star . . . 90

2.15 Prior and posterior distribution for the surface plus curvature energy per surface nucleon versus isospin asymmetry . . . 91 2.16 Correlation of the crust-core transition density and pressure with the

equation of state parameters for different filters . . . 93 2.17 Marginalized probability distributions for the crust-core transition

den-sity and pressure for different filters . . . 95 2.18 Correlation of the crust thickness and fractional crust moment of

iner-tia of a 1.4M⊙ neutron star with the equation of state parameters for

different filters . . . 97 2.19 Marginalized probabilities for the crust thickness and fraction of crust

moment of inertia of a 1.4M⊙ neutron star . . . 99

2.20 Marginalized posterior for the neutron star mass and radius of Vela . . 100 2.21 Marginalized posterior for the fraction of crust moment of inertia as a

function of the neutron star mass . . . 101 3.1 Free energy density difference between liquid and solid phases versus

temperature . . . 111 3.2 Crystallization temperature versus pressure for the one-component plasma

in the outer crust . . . 121 3.3 Equilibrium OCP composition versus baryon density in the outer crust

at finite temperature . . . 123 3.4 Normalized probability distribution of the atomic number Z in the

outer-crust regime . . . 124 3.5 Equilibrium composition of the multicomponent plasma versus pressure

in the outer-crust regime . . . 125 3.6 Impurity parameter versus pressure in the outer-crust regime . . . 127 3.7 Fraction of odd-A and odd-Z nuclei versus pressure in the outer crust

regime . . . 128 3.8 Equilbrium value of proton number versus baryon density in the inner

crust for BSk14 at finite temperature . . . 131 3.9 Free energy density versus proton number Z for different

thermody-namic conditions in the free neutron regime . . . 132 3.10 Crystallization temperature and equilibrium value of Z of the

3.11 Crystallization temperature and equilibrium value of Z of the one-component plasma versus baryon density in the inner crust . . . 135 3.12 Equilibrium proton fraction of the one-component plasma versus

tem-perature in the inner-crust regime . . . 136 3.13 Equilibrium composition of the multicomponent plasma versus baryon

density in the inner-crust regime . . . 137 3.14 Normalized probability distribution of the atomic number Z in the

1.1 Ground state of the outer crust . . . 16 1.2 Ground state of matter at the neutron-drip point . . . 19 1.3 Empirical parameters for SLy4, BSk24, BSk22, and DD-MEδ, and

ex-tracted from nuclear experiments . . . 23 2.1 Empirical parameters for BSk14, PKDD, and TM1 . . . 60 2.2 Minimum value and maximum value of each of the empirical parameters

for the prior distribution . . . 73 2.3 68% confidence intervals for the crust-core transition density and

pres-sure for different filters . . . 94 2.4 68% confidence intervals for the crust thickness and fractional crust

moment of inertia of 1.4M⊙ neutron star for different filters . . . 98

3.1 Value of the fraction and baryon mass fraction of odd nuclei in the outer crust for three selected temperatures . . . 129

Astrophysical context

The existence of dense stars was hypothesized as early as 1931 by Landau [Lan32], a year before the discovery of neutrons by Chadwick [Cha32]. Landau anticipated that the stellar matter density may become “so great that atomic nuclei come in close contact, forming one gigantic nucleus”. In 1934, Baade and Zwicky introduced the term supernova (SN) to designate a “remarkable type of giant novae”, a rare and very energetic phenomenon, characterized by a sudden and ephemeral burst in luminosity followed by a slow decay, and they predicted that “supernovae represent the transitions from ordinary stars to neutron stars” [BZ34]. The presence of neutron stars (NS) in the Universe remained purely theoretical until 1968, when a rapidly pulsating source, a

pulsar, was observed for the first time by Jocelyn Bell, a graduate student supervised

by Antony Hewish [Hew+68]. Several weeks after this observation, and motivated by the discovery of the Crab pulsar in 1968 which could not be identified as a white dwarf on account of a very short pulsation period [Com+69], pulsars were identified as “rotating neutron stars” by Gold [Gol68], paving the way for important theoretical development and observations in the following decades.

Formation and structure of neutron stars

During their life, stars of mass greater than ∼ 10M⊙ (M⊙ being the mass of the Sun),

can ignite their core elements up to silicon burning into iron, then the fusion of ele-ments is no longer possible since iron is the most stable nuclide in nature. The chain of reactions in their core ends, and in their final stage the stars exhibit an onion-like struc-ture, their core being composed of iron and neutron-rich iron-group nuclei [Bet+79], surrounded by shells of lower and lower burning elements up to possible inert hydro-gen, at progressively lower temperatures and densities [WHW02]. At this point, the stratified core is essentially sustained by the electron degeneracy pressure, and its mass keeps increasing through accretion as silicon shells are consumed, until it overtakes the Chandrashekar mass limit, MCh ∼ 1.44M⊙, when the gravitational force overcomes

the electron degeneracy pressure [Cha31], which has the effect of triggering a core-collapse supernova (CCSN) explosion [Jan+07]. In the aftermath of a CCSN, a warm protoneutron star (PNS) is left behind, with temperatures exceeding 1010 _{K, and two}

outcomes are possible: either the PNS will end up as a cold NS, or into a black hole if its mass is larger than the NS maximum mass, which is uncertain up to the present. Few minutes after the formation of the hot PNS, it transforms into an ordinary NS

Figure 1: Schematic representation of the inside of an NS. Figure taken from [PR06].

which is transparent for neutrinos.

The star cools down by emitting neutrinos and photons, and at approximately 108 _{K matter is catalyzed, that is in its ground state. A schematic representation of}

crust is generally subdivided into two regions: the outer crust and the inner crust, and the border between them is situated at the neutron drip surface, a few hundred meters from the atmosphere bottom. Inside the crust, atoms are fully ionized and form a lattice immersed in a relativistic electron gas, and if the neutron chemical potential is larger than the neutron rest mass, additionally in a neutron gas. Due to electron capture, matter is more neutron-rich with increasing density, further toward the interior of the star. In the bottom layers of the inner crust, nuclei are expected to exhibit nonspherical shapes. As the temperature of the inner crust falls below the critical temperature Tc ∼ 1010K, free neutrons with anti-aligned spins and zero orbital angular momentum form Cooper pairs and behave in a superfluid state, characterized by the absence of viscosity. At about half the saturation density nsat, corresponding to the equilibrium density of symmetric homogeneous nuclear matter, the crust-core interface is reached and nuclei disappear. Once again, we can distinguish the outer core, corresponding to the density range 0.5nsat . nB . 2nsat, and the inner core, where nB & 2nsat (nB being the baryon density). In the outer core, matter consists of a mixture of neutrons, protons, electrons, and possibly muons. The composition of the inner core is however uncertain, and several hypotheses have been put forward, such as the appearance of hyperons, boson condensates, and/or a phase transition to quark matter.

Observables of (proto)neutron stars

Pulsars were identified to rotating NS which produce pulsed emission, soon after their chance discovery in 1967 by Jocelyn Bell [Hew+68]. Five decades later, we have now observed about 3000 of them, and numerous techniques have been developed to measure their characteristic observables.

Global properties

It is easier for astronomers to measure the mass of an NS belonging to a binary system. There are several types of binaries: x-ray binaries, double NS binaries, radio pulsar–white dwarf binaries, and pulsars in binaries with nondegenerate stars (main-sequence stars). Depending on the type of binary, different techniques are used to infer the NS mass [HPY07]. For example, in double NS binaries the relativistic Shapiro delay [Sha64] can be exploited to infer the mass via pulsar timing, which consists in the regular monitoring of the rotation of a pulsar over long time periods. The relativistic Shapiro delay is a phenomenon from which precise masses for both a millisecond pulsar and its companion can be inferred [Dem+10;Cro+20]. Let us notice however that it is only observed in a small subset of high-precision, highly inclined binary pulsar systems. Measured NS masses via pulsar timing are displayed in Fig. 2.

Figure 2: Masses measured from pulsar timing. Vertical dashed (dotted) lines indicate

category error-weighted (unweighted) averages. Figure taken from [Lat19].

(LMXBs) during periods of little to no accretion, called quiescence [BBR98]. A num-ber of quiescent LMXBs have been studied with the Chandra and/or XMM–Newton observatories [Hei+14;Ser+12;GR14; Gui+13]. However, the results strongly depend on the assumptions made on the composition of the neutron star atmosphere which is poorly known [Ste+18]. An important improvement is expected from the analy-sis of the recent NICER mission, whose first results start to be available [Bog+19a;

Bog+19b;Mil+19;Raa+19;Ril+19], even if complications in the interpretation of the data arise due to the nonuniformity of the temperature over the surface (hot spots) [Bog+19a; Bog+19b; Mil+19; Raa+19; Ril+19]. Typical values for the NS masses and radii are M = 1.4M⊙ and R = 10 − 14 km, respectively.

de-formability of NS [Abb+18]. The tidal deformability describes how much a body is deformed by tidal forces, which arise when two massive bodies are in orbit around each other. The simplest and best known example corresponds to the Moon causing the tides observed in Earth’s oceans. The detection of gravitational radiation emitted by inspiraling binary NS is possible using ground-based GW detectors such as LIGO and Virgo. Shortly before merging, once the relative distance between the stars is small enough, the tidal distortion of the NS become so large that, in some cases (in strongest signals corresponding to closest merging events), it becomes possible to infer the tidal deformability from the GW signal.

Pulsar glitches

The so-called pulsar glitches are sudden jumps in the rotational frequency of a compact star. They are thought to originate from an abrupt transfer of angular momentum from the superfluid components of the NS, acting as an angular momentum reservoir, to the solid crust of the star, and all the normal fluid components which are strongly coupled to the crust by mutual dissipation. This sudden transfer is thought to be due to the unpinning of the superfluid vortices from the crystal lattice [AI75]. Indeed, a rotating superfluid, such as the superfluid neutrons in the inner crust of the NS, produces individual quantized vortices, with a density proportional to the rotational rate. Those vortices migrate towards the surface of the star due to quantization of the macroscopic vorticity, where they get pinned to the ions of the lattice that constitutes the solid crust of the NS. Since the star experiences a spin-down due to the emission of electromagnetic radiation, a differential lag develops between the faster superfluid vortices and the slower crust, leading to crustal stress. When the differential lag between the slower solid crust and faster superfluid vortices reach some threshold and can no longer be sustained, the vortices suddenly unpin from the lattice sites, leading to an angular momentum transfer to the crust, and the rest of the star which is entangled with the crust by mutual friction, so as to recover a close equilibrium between the normal and superfluid components. Since the electromagnetic slowing down is a continuous process, this is not a final equilibrium situation, and eventually stresses start to build up again, ultimately leading to another glitch event. At the time of writing, 555 glitches have been observed in 190 pulsars through high-precision pulsar timing [Esp+11; Jod]. The Vela pulsar (PSR B0833-45) is one of the most active glitchers known, with glitches occurring four times per decade on average.

Microphysics of the (proto)neutron star crust

regime cannot be exactly solved, the EoS is strongly model dependent, which induces considerable uncertainties in the prediction of astrophysical observables. In this the-sis, we are particularly interested in the modeling of the (P)NS crust, where matter is inhomogeneous. From the modeling point of view, the regime of subsaturation matter constitutes a challenging part of the nuclear EoS. Indeed, the uncertainties do not only concern the nuclear energy functional, but also the many-body method used to model inhomogeneous matter. Indeed, the evaluation of the EoS implies to know the microscopic composition at each point in the star. At finite temperature, an extra complication arises from the statistical mechanics treatment of the problem. Histori-cally, the stellar EoS at subsaturation densities was first calculated within the so-called single-nucleus approximation (SNA) [BBP71; NV73], based on the assumption that matter can be represented by the most probable nucleus given by the minimization of the free energy density of matter. While this approximation is exact at zero tempera-ture, a full distribution of clusters should be considered at finite temperatempera-ture, as is it the case in nuclear statistical equilibrium (NSE) models. Again, the exact solution of the many-body problem at finite temperature being out of scope, modeling cannot be avoided, which induces a model dependence on the calculation of observables.

One could naively consider that an optimal nuclear model can be extracted by confronting the theoretical predictions with the observational data. However, a major issue when considering characteristic EoS is that some observables are better repro-duced by a particular model (or a class of models), while they fail to reproduce other observables. In addition, each observable is associated with uncertainties and the ability to reproduce or not a determined measurement does not have the same im-pact on the reliability of the model, depending on which observable is considered. In addition to astrophysical constraints, there are also constraints coming from nuclear experiments, and recent developments in ab initio calculations based on chiral effec-tive field theory (EFT) [DHS16], which consists in the determination of the nuclear energy functional from a systematic power expansion which respects the fundamen-tal symmetries of low-energy QCD [ME11], that is the theory of strong interactions. The different models that can be considered have not been confronted to all those constraints, and not on the same level. In these circumstances, it is very arduous to validate (or invalidate) a model. A solution for this impasse is provided by the use of Bayesian inference principle, which allows to update our prior beliefs on the EoS with the constraints arising from the multiple sources mentioned above. It has been shown that the (P)NS observables are sensitive to the microphysics entering the EoS, for instance to the high-order derivatives of the nuclear symmetry energy and to the surface properties of finite nuclei. It is therefore essential to constrain these inputs in order to control the uncertainties in the observables. In this thesis, we are interested in making realistic predictions and to investigate the sources of uncertainties in the observables of nonaccreting cold NS and warm PNS, using the present day constraints provided by nuclear experiments, theoretical developments in chiral EFT, and astro-physical observations. This general argument applies to the total modeling of the NS, and also to the NS crust, which is the main focus of this work.

and cooling. In addition to the nuclear EoS, the determination of crustal observables requires the knowledge of the density and pressure at the transition point from the solid crust to the liquid core [PFH14]. In order to validate the full crustal origin of the large glitches observed in some pulsars, such as Vela, the neutron star crust must be sufficiently thick to store a significant amount of angular momentum. The corre-sponding fraction of the crust moment of inertia can be estimated [LEL99; And+12;

Del+16] in a range going from 1.6% up to 15%, depending on the importance of the effect of crustal entrainment, which is currently under debate [MU16; WP17]. A reli-able estimation of the crust thickness and of the associated moment of inertia is thus crucially needed to validate the crustal origin of pulsar glitches. For all these appli-cations, it is essential to have objective criteria allowing to validate or invalidate the different models, and possibly correlate the residual uncertainty of model predictions to well-defined parameters that can be constrained in the future by more precise ex-periments or ab initio calculations. The first part of the present thesis is aimed at providing a step forward in this direction. This will be done by introducing a flexible metamodeling procedure which will allow us to confront a very large set of models of catalyzed nuclear matter to the different constraints coming from both low-energy nuclear physics and astrophysical observations of mature NS.

The second part of the thesis will deal with the modeling of the crust at finite tem-perature. Again, the focus will be put on the determination of reliable error bars on the determination of astrophysical quantities due to the uncertainties of the modeling. This finite temperature modeling is not only essential to describe PNS dynamics, but it might also be relevant for the description of crustal observables of mature neutron stars. Indeed, the crust of an NS is unlikely to be in full thermodynamic equilibrium at zero temperature: NS are born hot, and if their core cools down sufficiently rapidly, the composition might be frozen at a finite temperature [Gor+11]. Deviations from the ground-state composition in the cooled crust around the neutron-drip density were al-ready considered in [BC79], but simple extrapolations of semiempirical mass formulae were used at that time. The value of the freeze-out temperature is difficult to evaluate, but a lower limit is given by the crystallization temperature, since we can expect that nuclear reactions will be fully inhibited in a Coulomb crystal. For these reasons, the last part of this thesis will be dedicated to the study of the structure of the crust at the temperature of crystallization. The possible presence of amorphous and heterogeneous phases in the inner crust of a neutron star is expected to reduce the electrical conduc-tivity of the crust, with potentially important consequences on the magneto-thermal evolution of the star. The study of cooling processes is important for comparing the theoretical calculations with surface temperature as measured by satellites. In cool-ing simulations, the disorder is quantified by an impurity parameter which is often taken as a free parameter. This parameter is directly related to the electron–impurity conductivity, which contributes to the total thermal conductivity [FI76].

Structure of the thesis

Structure and equation of state of

cold nonaccreting neutron stars

This chapter deals with the structure and EoS of NS within the framework of the “cold catalyzed matter” (CCM) hypothesis.

With typical temperature of about T ∼ 108 _{K ∼ 0.01 MeV, NS are very cold}

systems from the nuclear physics viewpoint, therefore the CCM hypothesis is com-monly used to predict their internal composition and pressure. In this limit, thermal, nuclear, and beta equilibrium are established at zero temperature, meaning that the energy cannot be lowered by weak, strong, or electromagnetic processes, thus the mat-ter is in its ground state. It is reasonable to expect these equilibrium conditions to be valid in any NS as far as it is not accreting matter from a neighbor. Indeed, in the accretion scenario, the typical timescale of the process are such that the matter composition is believed to be out of equilibrium.

As already discussed in the general introduction, the evaluation of the EoS implies to know the microscopic composition at each point in the star. At subsaturation densi-ties, the solid crust consists mainly of clusterized matter, arranged in a body-centered cubic lattice [HPY07]. The relevant degrees of freedom in the crust are the Wigner-Seitz (WS) cells, containing exactly one lattice point [WS33]. At zero temperature, WS cells are supposed to be identical, thus the SNA, which considers a unique con-figuration for a given thermodynamic condition of temperature and pressure (P, T ), becomes exact. The ground state of the outer crust is almost entirely characterized by experimental nuclear masses, which are available up to (N − Z)/A . 0.3, N being the number of neutrons, Z the number of protons, and A = N + Z the number of nucleons constituting atomic nuclei. The determination of inner-crust ground state is however more challenging because the crust is permeated by free neutrons, a situation which cannot be achieved in terrestrial conditions. Therefore, different treatments, from microscopic [NV73] to classical [BBP71], can be envisaged to estimate the energy of matter, and the EoS in this region depends on the nucleon-nucleon (NN) effective interaction or energy functional. At suprasaturation densities, matter consists of a uniform plasma of neutrons, protons, electrons, and eventually muons, in both strong and weak equilibrium. The development of a unified EoS, that is such that matter at subsaturation and supersaturation densities are treated within a unique model, is

essential if one wants to make realistic predictions on NS observables [For+16]. The plan of the chapter is as follows. In Section 1.1, the ground state of the outer crust is determined by application of the method introduced by Baym, Pethick, and Sutherland (BPS) [BPS71], using experimental masses supplemented by state-of-the-art microscopic theoretical mass tables. Section 1.2 is devoted to the determination of the inner-crust ground state using a compressible liquid drop model (CLDM) based on the metamodeling technique [MHG18a; CGM19a]. The phase transition from the solid crust to the liquid core, occuring at some ≈ 1 km from the surface of the star, is investigated. In Section 1.3, we calculate the ground state of matter in the outer core, and we address the problem of the inner-core composition. In Section 1.4, a unified metamodeling of the EoS is proposed. Finally, conclusions are given in Section 1.5.

Contents

1.1 Ground state of the outer crust . . . . 11 1.1.1 Wigner-Seitz cell energetics . . . 11 1.1.1.1 Nuclear mass tables . . . 12 1.1.1.2 Relativistic electron gas . . . 12 1.1.2 The BPS model . . . 13 1.1.3 Equilibrium composition and equation of state . . . 15 1.2 Ground state of the inner crust . . . . 19 1.2.1 Modeling the nuclear energy . . . 20 1.2.1.1 Metamodeling of homogeneous nuclear matter . . 22 1.2.1.2 From homogeneous nuclear matter to finite nuclei

1.1

Ground state of the outer crust

At zero temperature, the matter inside the outer crust corresponds to a lattice of strongly bound nuclei, immersed in a sea of electrons. The mass density at which nuclei are fully ionized and electron completely degenerated is of the order of ρB ≫ 6AZ ∼ 104 _g/cm3 _for 56_{Fe, which is the ground state of matter at very low density.}

Below 104 _g/cm3_{, some electrons are still bound to the nuclei and one must rely on}

the EoS calculated by Feynman, Metropolis, and Teller from 15 to 104 _g/cm3_{, suitable}

for the envelope of neutron stars [FMT49].

This section deals with the determination of the outer-crust ground state. In 1.1.1, we detail the different terms entering the WS cell energy, with emphasis on the rela-tivistic electron gas energy as well as nuclear masses. The ground state of the outer crust is calculated by application of the variational BPS method [BPS71], which is presented in 1.1.2. Finally, using current knowledge on experimental masses [Hua+17] supplemented by different microscopic theoretical mass tables, we compute the equi-librium composition and EoS. Our results are presented in 1.1.3.

1.1.1

Wigner-Seitz cell energetics

In the outer crust, a WS cell is composed of a strongly bound nucleus at the center, immersed in a relativistic electron gas of density ne. Charge neutrality is assured in each unit cell, ne = np, with np the proton density inside the cell.

The energy of a WS cell in the outer crust can therefore be written as

EW S = Ei+ VW Sεe, (1.1)

with Ei the ion energy, VW S the volume of the cell, and εe the energy density of the free electron Fermi gas. The ion energy reads

Ei = M′(A, Z)c2+ EL+ Ezp, (1.2) where M′_{(A, Z)c}2 _{is the nuclear mass of a nucleus with associated mass number A}

and charge number Z, EL the temperature-independent static-lattice term, and Ezp the zero-point quantum vibration term given by

Ezp = 3

2~ωpu1, (1.3)

where u1 = 0.5113875 is a numerical constant for a body-centered cubic lattice (see

Table 2.4 of [HPY07]), which is assumed to be the geometry minimizing the lattice energy. This assumption was recently confirmed in [CF16]. The ion plasma frequency

ωp is given by ~_ωp = v u u t(~c) 2_4πn N(Ze)2 M′_{(A, Z)c}2 , (1.4)

e being the elementary charge, c the speed of light, ~ = h/2π the reduced Planck

constant, and where the ion density nN = 1/VW S has been introduced. The lattice energy reads

EL= −CM (Ze)2

aN

with CM = 0.895929255682 the Mandelung constant for a body-centered cubic lattice (see Table 2.4 of [HPY07]), and aN = (4πnN/3)−1/3 the ion-sphere radius.

1.1.1.1 Nuclear mass tables

An essential input for Eq. (1.2) is the nuclear mass table. When available, that is for I = (N − Z)/A . 0.3, we use experimental masses from the 2016 Atomic Mass Evaluation (AME) [Hua+17]. For more neutron-rich nuclei and until we reach the neutron drip line, the use of a model is required, thus a model dependence arises. A possibility is to rely on microscopic Hartree-Fock-Bogoliubov (HFB) theoretical mass tables [Sam+02], which are based on the nuclear energy-density functional theory.

In general, atomic masses are tabulated instead of the nuclear ones which can be calculated as

M′(A, Z)c2 = M(A, Z)c2− Zmec2+ Be(Z), (1.6) with M(A, Z)c2 _{= ∆ǫ + Am}

uc2 the atomic mass (∆ǫ is the mass excess and mu is the atomic mass unit), me the electron mass, and Be the binding energy of atomic electrons depending solely on the number of protons Z according to the approximation proposed in [LPT03] (see their Eq. (A4)),

Be(Z) = 1.44381 × 10−5Z2.39+ 1.55468 × 10−12Z5.35. (1.7)

1.1.1.2 Relativistic electron gas

In the cold outer crust, the mass densities are above ∼ 104 _g/cm3 _{therefore the}

elec-trons are essentially free. In this regime, it was shown in [WI03] that electron-charge screening effects are negligible and that the electron density is essentially homoge-neous. This is explained from the fact that the electron Thomas-Fermi screening length is larger than the lattice spacing. The expression of the energy density of a relativistic electron gas, with rest mass energy, at zero temperature can be calculated as εe(ne) = Z ke 0 k2_dk π2 c q ~2_k2_{+ m} e2c2 = Pr 8π2 h xr(1 + 2xr2)γr− ln(xr+ γr) i , (1.8)

with Pr = me4c5/~3 the relativistic unit of the electron pressure, xr = ~ke/(mec) the relativity parameter, and γr =

√

1 + xr2. ke is the electron Fermi wave number given by

ke = (3π2ne)1/3. (1.9)

The derivation of Eq. (1.8) is given in Appendix A. Above 107 _g/cm3_{, x}

r ≫ 1, thus electrons can be considered ultrarelativistic and Eq. (1.8) becomes

εe(ne) = 3

Taking the derivative of Eq. (1.8) with respect to the electron gas density yields the pressure, Pe = − ∂(VW Sεe) ∂VW S = ne ∂εe ∂ne − εe = Pr 8π2  xr 2 3x 2 r− 1  γr+ ln(xr+ γr)  . (1.11)

The electron exchange energy, which is a direct consequence of the Pauli exclusion principle, and the electron correlation energy, which is due to the fact that the motion of each electron is affected by the motion and position of the other electrons, are neglected since they are known to be small in comparison with the kinetic energy of relativistic electrons. The expression of the electron exchange correction to the free energy density for a strongly degenerate electron gas is given in [HPY07] (see their Eq. (2.151)). We have checked that the inclusion of these corrections does not modify the results presented in this chapter.

1.1.2

The BPS model

The variational technique which is currently used to calculate the ground state of the outer crust was introduced by Baym, Pethick, and Sutherland in [BPS71].

The thermodynamic potential to be minimized is the zero-temperature Gibbs free energy per nucleon at fixed pressure P under the condition of charge neutrality ne =

ZnN, until the neutron drip sets in, the condition for which is µn= mnc2, with µnthe neutron chemical potential, and mn the neutron mass.

The definition of the zero-temperature Gibbs free energy per nucleon is

g = εW S + P nB

, (1.12)

where εW S = EW S/VW S is the energy density of the WS cell, and nB is the baryon density given by nB = nNA = A/VW S. We can calculate the pressure as

P = nB2 ∂(εW S/nB) ∂nB     Z,A , (1.13)

yielding, using Eq. (1.1),

P = Pe+ 1

3ELnN + 1

2EzpnN. (1.14)

Thus the expression of the zero-temperature Gibbs free energy per nucleon can be rewritten as g = M′(A, Z)c2+ 4 3EL+ 1 2Ezp+ Zµe A , (1.15)

where µe is the electron chemical potential given by

µe=

∂εe

∂ne

Essentially, one fixes the pressure P , calculates for each nucleus (A, Z) the electronic density by solving numerically Eq. (1.14), then constructs a table g(A, Z). The ground state of the outer crust at pressure P then corresponds to the nucleus associated to the minimum value of g.

As explained in [Pea+18], the neutron chemical potential can be calculated through

µn= g, (1.17)

because the beta equilibrium equation µn = µp+µe holds throughout the star. Indeed, by exploiting the thermodynamic relation

G = ε + P = µnnn+ µpnp+ µene, (1.18) together with the charge neutrality condition ne = np, we obtain

G = µnnn+ (µe+ µp)np, (1.19) where G is the zero-temperature Gibbs free energy density, and µp the proton chemical potential. In the case where the chemical equilibrium is established, that is if weak processes are at equilibrium, we finally get

G = µnnB. (1.20)

One should stress that while minimizing the zero-temperature Gibbs free energy per nucleon at fixed pressure is less practical than simply minimizing the total energy density at constant baryon density, it makes it easier to study the transitions between layers. The pressure increases continuously with increasing depth in the star, thus a discontinuity in the density is the signature of a transition from a layer (A, Z) to another (A′_{, Z}′_{). Noting that the pressure in the outer crust is approximately equal}

to the pressure of the electron gas (the lattice and zero-point terms contribute to less than 5% to the total pressure in the bottom layers of the outer crust), we know that

ne= np is continuous across the transition thus ∆nB = n′B− nB ≃ np A′ Z′ − A Z ! , (1.21)

and the fractional change in the baryon mass density results ∆ρB ρB ≃ ∆nB nB ≃ Z/A Z′_/A′ − 1. (1.22)

For this reason, it is more convenient to choose the pressure as the independent vari-able. In this way we avoid to make a Maxwell construction to estimate the pressure at which the transition from (A, Z) to (A′_{, Z}′_{) occurs.}

been made to measure nuclear masses near the neutron drip line [LPT03], as well as theoretical developments were achieved to construct microscopic mass tables [Sam+02;

GCP13]. It is therefore important to reevaluate the ground state of the outer crust considering those experimental and theoretical advances. In this line, Haensel and Pichon studied the consequences of progress concerning the experimental determina-tion of atomic masses in 1994 [HP94]. The authors found that the ground state of the outer crust can be determined exclusively by experimental masses in a fully model independent way up to ρB ≈ 1011 g/cm3. From this density up to the neutron drip point, the phenomenological, liquid-drop based mass formula of Möller was used, the formalism of which is described in [MN88].

In the same spirit as [HP94], we calculate, in the following, the ground state of the outer crust using the present day knowledge on experimental masses of neutron rich nuclei [Hua+17;Wel+17] combined with state-of-the-art microscopic theoretical mass tables [GCP13].

1.1.3

Equilibrium composition and equation of state

We turn to the numerical results obtained applying the BPS method presented in 1.1.2. The ground state of the outer crust is calculated, beginning at P = 3×10−11_MeV/fm3_,

corresponding approximately to nB ≈ 10−9 fm−3, in order to ensure complete ioniza-tion and electron degeneracy. The pressure is increased from steps of 0.02P until the neutron drip point, defined by µn− mnc2 = 0, is reached.

The ground-state composition and EoS of the outer crust of a cold nonaccreting NS is reported in Table 1.1. The upper part of the table, nB < 3.84 × 10−5 fm−3, corresponding to ρB <6.39 × 1010 g/cm3, is exclusively determined by experimental data from the AME2016 [Hua+17]. Let us notice that the maximum mass density at which experimentally studied nucleus 80_{Zn was present was found to be slightly lower}

in [HP94], ρB = 5.44 × 1010 g/cm3. This is due to the fact that the determination of this density depends on the mass formula used, here HFB-24, for the neutron rich nuclides present in the bottom layers of the outer crust. From 3.84×10−5 _fm−3_{, matter}

becomes so neutron rich that the nuclear masses cannot be measured experimentally for now, thus a model dependence is expected to arise because the nuclear masses have to be extrapolated from laboratory nuclei. The nuclear mass model used to calculated the ground state here is HFB-24 [GCP13], constructed from the BSk24 functional following the Hartree-Fock-Bogoliubov method [Sam+02]. The BSk24 functional is part of a family of functionals labeled BSk22 to BSk26, consisting of unconventional effective Skyrme forces with extra t4 and t5 terms that behave as density-dependent

generalizations of the usual t1and t2terms present in traditional Skyrme forces [VB72;

Cha+97]. The functionals BSk22-26 were fitted to the 2353 experimental masses of the AME2012 [Aud+12] and differ mainly by their symmetry energy S, defined here as the difference between the energy per nucleon of pure neutron matter (PNM) eP N M and the energy per nucleon of symmetric nuclear matter (SNM) eSN M,

element Z N Yp nB,max Pmax µn− mnc2 µe (fm−3_{) (MeV/fm}3_{) (MeV) (MeV)} 56_{Fe 26 30 0.4643 4.97 × 10}−9 _{3.40 × 10}−10 _{-8.96 0.95} 62_{Ni 28 34 0.4516 1.56 × 10}−7 _{4.09 × 10}−8 _{-8.26 2.57} 64_{Ni 28 36 0.4375 8.07 × 10}−7 _{3.60 × 10}−7 _{-7.52 4.34} 66_{Ni 28 38 0.4242 9.27 × 10}−7 _{4.16 × 10}−7 _{-7.46 4.50} 86_{Kr 36 50 0.4186 1.85 × 10}−6 _{1.03 × 10}−6 _{-7.01 5.63} 84_{Se 34 50 0.4048 6.85 × 10}−6 _{5.64 × 10}−6 _{-5.87 8.59} 82_{Ge 32 50 0.3902 1.67 × 10}−5 _{1.77 × 10}−5 _{-4.82 11.41} 80_{Zn 30 50 0.3750 3.84 × 10}−5 _{5.10 × 10}−5 _{-3.58 14.86} 78_{Ni 28 50 0.3590 6.68 × 10}−5 _{1.01 × 10}−4 _{-2.63 17.61} 126_{Ru 44 82 0.3492 7.52 × 10}−5 _{1.12 × 10}−4 _{-2.47 18.15} 124_{Mo 42 82 0.3387 1.21 × 10}−4 _{2.04 × 10}−4 _{-1.54 21.05} 122_{Zr 40 82 0.3279 1.56 × 10}−4 _{2.75 × 10}−4 _{-1.03 22.69} 121_{Y 39 82 0.3223 1.63 × 10}−4 _{2.84 × 10}−4 _{-0.98 22.86} 120_{Sr 38 82 0.3167 1.95 × 10}−4 _{3.52 × 10}−4 _{-0.60 24.12} 122_{Sr 38 84 0.3115 2.37 × 10}−4 _{4.49 × 10}−4 _{-0.15 25.62} 124_{Sr 38 86 0.3065 2.56 × 10}−4 _{4.87 × 10}−4 _{0.00 26.14}

Table 1.1: Ground state of the outer crust of a cold nonaccreting neutron star.

Exper-imental data from the 2016 Atomic Mass Evaluation [Hua+17] are used when available. Mass excesses of 77−79_{Cu are taken from [Wel+17]. Experimental masses are}

supple-mented with masses from microscopic HFB-24 theoretical mass table [GCP13] (lower part). The last line corresponds to the neutron drip point. The following quantities are reported in the table: the element, the atomic number Z, the neutron number N , the proton fraction Yp = Z/A (A is the number of nucleons), the maximum baryon

density nB,max at which the nuclide is found, the associated pressure Pmax, the neutron

chemical potential minus rest mass µn− mnc2, and the electron chemical potential µe.

with the so-called symmetry energy parameter Esym. This quantity is constrained to 32, 31, 30, and 29 MeV for BSk22, BSk23, BSk24, and BSk25, respectively. For BSk26,

Esym = 30 MeV as well but the functional is fitted to the APR EoS of PNM [APR98], unlike BSk22-25 that are fitted to the stiffer LS2 EoS [LS08].

The first layer of the outer crust consists of a crystal lattice of 56_{Fe, the mass per}

nucleon of which is the lowest among all nuclides. The sequence of nuclides is in good agreement with the results of [HP94]. One can note the persistence of magic numbers

find two additional layers of strontium (N = 84 and N = 86) with respect to [HP94] results, and the layer of 118_{Kr before the neutron drip is not observed.}

At high density, a thin layer of odd-mass nuclei 121_{Y is found with the HFB-24}

mass model. It is interesting since the possibility of having odd nuclei in the ground state of the outer crust was not considered in the original calculation of BPS [BPS71]. Also, it is reported in [Pea+18] that 79_{Cu is favored over} 76_{Ni, found for HFB-22 and}

HFB-25, in the case where the recent mass excess measurements of [Wel+17] are not included. This highlights the importance of measuring the mass of odd-nuclei. Indeed, one could expect that the presence of odd-nuclei in the outer crust of NS might lead to a ferromagnetic phase transition at low temperature, which would generate a magnetic field and alter the existing field, and so the electron gas.

It can also be observed in Table 1.1 that the proton fraction Yp = Z/A always de-creases with increasing depth. The neutronization of matter can be understood by the following reason. Neglecting the static-lattice energy as well as the zero-point quan-tum vibrations terms, we can write the zero-temperature Gibbs energy per nucleon as

g ≃ M ′_{(A, Z)c}2_{+ V ε} e nBV + P nB . (1.24)

Since the lattice does not contribute to the pressure, we have P ≃ Pe = neµe− εe, yielding g _≃ M ′_{(A, Z)c}2 A + ne nB µe. (1.25)

The charge neutrality is ensured in the unit cell, therefore ne/nB = Z/A and we finally get g _≃ M ′_{(A, Z)c}2 A + Z Aµe. (1.26)

For ρB≫ 10−7 g/cm3, electrons are ultrarelativistic and the electron chemical poten-tial is calculated by taking the derivative of Eq. (1.10),

µe = ∂εe ∂ne = 3 4~c(3π 2₎1/3_n1/3 e . (1.27)

Therefore, the electron chemical potential µe scales as P1/4 (P ∝ n4/3e ). Then it ap-pears that, with increasing pressure, it is energetically favorable to decrease the proton fraction Yp = Z/A in order to compensate the increase in the term M′(A, Z)c2/A. The neutron chemical potential monotonously increases with increasing density and at nB = 2.56 × 10−4 fm−3, corresponding to P = 4.87 × 10−4 MeV/fm3, the neutron drip is finally reached. It should be mentioned that the neutron drip density as well as pressure depend on the mass model, here chosen to be HFB-24.

10

10

n

[fm

]

30

40

50

60

70

80

90

N

Z

HFB-24

HFB-26

HFB-14

FRDM

Figure 1.1: Variation with baryon density nB of the equilibrium value of atomic

number Z and neutron number N in the bottom layers of the outer crust for four different models: HFB-24, HFB-26 [GCP13], HFB-14 [GSP07], and FRDM [Mol+95].

observed from nB ≈ 2.5 × 10−5 fm−3, where the experimental mass data are not avail-able. We recover similar sequences of nuclides with the different mass models, with the strong shell effect associated to N = 50 at low density and N = 82 in the bottom layers. A thin layer with N = 52 is also found for HFB-24 and HFB-26 just before the transition to N = 82. While this transition occurs in the vicinity of 8 × 10−5 _fm−3 _for

HFB-24, HFB-26, and FRDM, it is found to happen at approximately 4 × 10−5 _fm−3

for HFB-14. Different nuclides are found close to the neutron drip, depending on the model: 124_{Sr is found for HFB-24,} 126_{Sr for HFB-26,} 120_{Kr for HFB-14, and} 122_{Kr for}

FRDM. Let us note that most of the mass of the outer crust is actually concentrated in those densest layers. The neutron drip density nN D and pressure PN D slightly depend on the model as well. These values are reported in Table 1.2.

The variation of pressure with baryon density, namely the EoS, is shown in Fig. 1.2 for the four mass models. The selected models give the same value of pressure up to ≈ 2.5 × 10−5 _fm−3_{, as it is the case for the composition. At higher densities, we can}

model element Z N nN D (fm−3) PN D (MeV/fm3) HFB-24 124_{Sr 38 86 2.56 × 10}−4 _{4.87 × 10}−4

HFB-26 126_{Sr 38 88 2.62 × 10}−4 _{4.91 × 10}−4

HFB-14 120_{Kr 36 84 2.67 × 10}−4 _{5.01 × 10}−4

FRDM 122_{Kr 36 82 2.62 × 10}−4 _{4.99 × 10}−4

Table 1.2: Chemical element, atomic number Z, neutron number N , density nN D,

and pressure PN D at the neutron drip point for four different mass models: HFB-24,

HFB-26 [GCP13], HFB-14 [GSP07], and FRDM [Mol+95].

0.00000 0.00005 0.00010 0.00015 0.00020 0.00025

n

[fm

]

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

P

[MeV/fm

]

HFB-24

HFB-26

HFB-14

FRDM

Figure 1.2: Variation with baryon density nB of pressure P for four different models:

HFB-24, HFB-26 [GCP13], HFB-14 [GSP07], and the FRDM [Mol+95].

1.2

Ground state of the inner crust

0

20

r [fm]

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

n

[fm

−

3

]

total

r-cluster

r-gas

0

20

r [fm]

total

e-cluster

e-gas

Figure 1.3: Wood-Saxon density profiles, for arbitrary values, within a WS cell in

the regime of the inner crust. Blue dashed lines correspond to the cluster density with different decompositions between cluster and gas (left, r-cluster; right, e-cluster), and orange dotted lines to the gas density. The total density profile is represented in gray solid lines.

the inner-crust ground state is detailed in 1.2.2. The numerical method as well as results are presented afterwards, in 1.2.3. In 1.2.4 we add Strutinsky shell corrections on top of the CLD energy as an attempt to recover magic numbers in the free neutron regime. Nonspherical pasta phases in the bottom layers of the inner crust are consid-ered in 1.2.5. Finally, the phase transition from the solid crust to the liquid core is investigated in 1.2.6.

1.2.1

Modeling the nuclear energy

Once the neutron dripline is reached, neutrons start to drip out of nuclei but stay confined in the WS cell because of the gravitational pressure, whereas they would have been emitted in the laboratory. In the regime of the inner crust, we thus have, in each unit cell, a cluster immersed in an electron sea, and an ambient neutron gas.

protons in the unit cell. In the former, the cluster occupies a volume at the center of the cell and is surrounded by the neutron gas. Evidently, there is a thin region where cluster and gas overlap given the fact that a sharp interface would be unrealis-tic. The r-cluster representation naturally emerges in the local density approximation in density functional theory. Indeed, if the energy is expressed as a function of the local density, then the cluster corresponds to the dense part and the gas to the dilute one. This interpretation is used in most of the calculations at finite temperature to model SN, see for example the renowned Lattimer and Swesty EoS [LS91]. In the e-cluster representation, the gas penetrates the cluster. This interpretation appears spontaneously in single-particle developments. Indeed, in very neutron-rich clusters, beyond the neutron dripline, the bound states are occupied as well as the resonant and continuum states. All the unbound single-particle states that are occupied thus represent the neutron gas, characterized by a quasihomogeneous spatial distribution, the continuum wave functions being very similar to plane waves. The difference be-tween the two representations is illustrated in Fig. 1.3. The cluster and gas density profiles, described by Woods-Saxon profiles with arbitrary values, are plotted along the WS cell radius in the r-cluster representation (left) and e-cluster representation (right). Woods-Saxon profiles are known to give a good description of medium-mass and heavy nuclei density profiles, and are commonly used in Thomas-Fermi (TF) and extended Thomas-Fermi (ETF) calculations [Ons+08; Pea+18]. In both representa-tions, the total density profile (solid gray line) is the same, and so must therefore be the total energy in a TF or ETF framework. One can switch between the two representations using the simple geometric relations

Ae= A  1 − ng n0  , Ze= A 1 − ng,p n0,p ! , (1.28)

where Ae (Ze) is the number of nucleons (protons) in the e-cluster, A (Z) the number of nucleons (protons) in the r-cluster, ng (ng,p) the total (proton) gas density, and n0

(n0,p) the average total (proton) density inside the cluster. Since we do not consider

the possibility of proton drip in the inner crust, we have ng,p = 0, yielding Ze = Z and implying that the total gas density is equal to that of the neutron gas. Indeed, while free protons are expected at non-zero temperature, their presence at zero temperature remains uncertain [BBP71] and depends on the nuclear model. For some models, the proton drip could set in the very bottom layers of the crust [Pea+18]. However, if nonspherical shapes are considered then it is found that protons remain in the cluster for most models.

At zero temperature, we define the mass of the cluster in the e-cluster representa-tion as

Mi,ec2 = (A − Z)mnc2+ Zmpc2+ Ecl− Vcl(εg + ngmnc2), (1.29) where εg represents the energy density of the neutron gas of density ng, Vcl = A/n0

the volume of the cluster, and Ecl the energy of the cluster, which will be specified lower in this chapter. The WS cell energy EW S can be written as

EW S = Mi,ec2+ VW Sεe+ VW S(εg+ ngmnc2)

with VW S the volume of the cell. The total number of protons ZW S and nucleons AW S inside the WS cell are given respectively by

ZW S = A1 − I 2 and AW S = A  1 −ng n0  + ngVW S, (1.31) where the cluster asymmetry I = (N − Z)/A has been introduced. Let us notice that the number of protons inside the cell remains an integer number, as in the outer crust, unlike the number of neutrons that is noninteger because of the outside neutron gas.

As for the bottom layers of the outer crust, the determination of the ground state of the inner crust is model dependent. Therefore, the two necessary ingredients of the WS cell energy are the equation of state of PNM, and the energy of the cluster, which will be treated in the CLD approximation introduced by Baym, Bethe, and Pethick (BBP) in their pioneering work [BBP71].

1.2.1.1 Metamodeling of homogeneous nuclear matter

Parameter Unit N SLy4 BSk24 BSk22 DD-MEδ Average Uncertainty Esat MeV 0 -15.97 -16.05 -16.09 -16.12 -15.8 0.3 nsat fm−3 1 0.1595 0.1578 0.1578 0.1520 0.155 0.005 Ksat MeV 2 230 246 246 219 230 20 Qsat MeV 3 -363 -274.5 -276 -748 300 400 Zsat MeV 4 1587 1184 1190 3950 -500 1000 Esym MeV 0 32.01 30.00 32.00 32.35 32 2 Lsym MeV 1 46.0 46.4 68.5 52.8 60 15 Ksym MeV 2 -120 -38 13 -118 -100 100 Qsym MeV 3 521 711 563 846 0 400 Zsym MeV 4 -3197 -4031 -3174 -3545 -500 1000 m∗ sat/m 0.69 0.80 0.80 0.69 0.75 0.1 ∆m∗ sat/m -0.19 0.20 0.20 -0.17 0.1 0.1

Table 1.3: Value of each of the empirical parameters, associated unit, and derivative

order N for SLy4 [Cha+98], BSk24, BSk22 [GCP13], and DD-MEδ [Roc+11] func-tionals. Average values and uncertainties extracted from experimental analysis are taken from [MHG18a].

and Zsym), respectively. The symmetry energy is generally defined as

esym_HM(n) = 1 2 ∂2_e HM(n, δ) ∂δ2      δ=0 , (1.32)

where eHM is the energy per nucleon in nuclear matter. In Table 1.3 are listed the value of each empirical parameter for the Skyrme-type SLy4 [Cha+98], BSk22, and BSk24 [GCP13], and relativistic DD-MEδ [Roc+11] functionals. Average values and associated uncertainties extracted from experimental analysis are also provided. It can be seen that the isovector parameters are in general less known than the isoscalar ones for the same derivative order. For instance, the relative uncertainty on Ksym is about 100% while that of Ksat is less than 10%. It should also be stressed that the high-order parameters Qsat(sym) and Zsat(sym) are poorly constrained by nuclear experiments.

The empirical parameters can be used in a Taylor expansion to estimate the nuclear matter EoS analytically up to n ≈ 2 − 3nsat [MHG18a; MHG18b]. Limiting us to derivative order N = 2, we obtain

eN =2_HM(n, δ) = Esat+ 1 2Ksatx 2_{+ δ}2_(E sym+ Lsymx+ 1 2Ksymx 2_{), (1.33)}

In a mean field approach, we can treat nucleons as independent particles, thus the kinetic part of the energy per particle of nuclear matter is given by the same functional form as a nonrelativistic Fermi gas (FG), that is

tF G_HM(n, δ) = 1 2tF Gsat(1 + 3x)2/3 " (1 + δ)5/3 m m∗ n + (1 − δ)5/3 m m∗ p # , (1.34) where tF G sat = 3~2/(10m)(3π2/2)2/3n 2/3

sat is the kinetic energy of SNM at the saturation density, m = (mn + mp)/2 is the mean nucleon mass, and m∗n (m∗p) is the neutron (proton) effective mass. The Landau effective mass is introduced in order to take into account the in-medium nuclear interaction that alters the mass of nucleons. It can be parametrized as (q = n, p) m m∗ q = X1 α=0 mq,α(δ) xα α! = 1 + (κsat+ τ3κsymδ)(1 + 3x), (1.35) with τ3 = 1 (τ3 = −1) for neutrons (protons). Two additional parameters κsat and

κsym, related to the effective mass m∗sat and isospin splitting ∆m∗sat at saturation density, are then introduced. They are defined at n = nsat as

κsat = m m∗ sat − 1, (1.36) κsym = 1 2 m m∗ n − m m∗ p ! . (1.37)

In principle, it is possible to reproduce any EoS model with a Taylor expansion, considering an infinite number of parameters, N → ∞, however the convergence would be very slow. In order to fasten the series convergence, one can add extra functional dependencies, which correspond to the true EoS in the limit of simplistic cases, but which allow, by judicious choices of empirical parameters, to reproduce with precision realistic functionals. With this observation in mind, a δ5/3 _{dependence is added as in}

Eq. (1.33), by decomposing the energy per particle into a potential and kinetic part, yielding

eN_HM(x, δ) = tF G

HM(n, δ) + vNM M(n, δ), (1.38) where vN

M M is the potential energy per particle expressed as a Taylor expansion in the parameter x at n = nsat, v_{M M}N (n, δ) = N X α≥0 (vis α + δ2vivα) xα α!. (1.39)

The quadratic approximation for the isospin dependence of the potential energy has been made in Eq. (1.39), as suggested by microscopic calculations in [Vid+09]. The coefficients vis

α and vαiv are mapped to the empirical parameters following